[Paper Review] Zitterbewegung and the Charge of an Electron

Basil S. Davis|arXiv (Cornell University)|Jun 25, 2020

Quantum Mechanics and Applications52 references4 citations

TL;DR

This paper reinterprets zitterbewegung in the Dirac equation as a rapid oscillation of the electron's charge rather than its mass, resolving the apparent contradiction of electron velocity eigenvalues at ±c. It shows that charge zitterbewegung arises from superpositions of positive and negative energy states, with measurable effects in strong fields, and argues that zitterbewegung cannot account for rest mass energy, challenging prior theories based on massive motion.

ABSTRACT

Dirac's Relativistic Wave Equation implies a measured electron velocity of $\pm c$ in any direction, in contradiction to Special Relativity and observation. It is shown in this article that this anomalous electron velocity reveals an internal structure of the electron whereby the mass and the charge of the electron cannot be described by the same position operator. The measured velocity of electron mass is always less than $c$ in any direction but charge can be displaced at the speed of light. This speed is realizable only when the electron is in a state that is a superposition of positive and negative energy states, also known as a zitterbewegung state. It is shown that in zitterbewegung it is the charge and not the mass that undergoes rapid spatial oscillation, and that there are measurable consequences of this charge zitterbewegung. Zitterbewegung of charge also occurs in an entangled electron-positron pair created by a strong electric field.

Motivation & Objective

• To resolve the paradox of electron velocity eigenvalues at ±c in the Dirac equation, contradicting special relativity.
• To clarify the physical interpretation of the velocity operator $ c\boldsymbol{\alpha} $, showing it applies to charge, not mass.
• To demonstrate that zitterbewegung is a real motion of the electron's charge, not its mass, with measurable consequences in supercritical electric fields.
• To challenge existing theories that attribute rest mass energy to zitterbewegung motion of the electron's mass.
• To establish that the electron's charge, not its mass, undergoes high-frequency oscillation at the speed of light in zitterbewegung states.

Proposed method

• Derives the velocity operator $ \dot{x}_i = c\alpha_i $ from the commutator $ i\hbar\dot{x}_i = -[H, x_i] $, showing eigenvalues ±c.
• Analyzes the Dirac Hamiltonian $ H = c\boldsymbol{\alpha}\cdot(\mathbf{p}-e\mathbf{A}) + \beta mc^2 + e\phi $ to identify the role of $ \alpha_i $ matrices in defining charge velocity.
• Applies field-theoretic modeling to the supercritical electric field limit $ W \to 0 $, showing zitterbewegung amplitude vanishes as field strength increases.
• Distinguishes between charge zitterbewegung (at speed c) and hypothetical mass zitterbewegung (not at speed c), using the principle of equivalence and gravitational coupling.
• Uses the position operator decomposition to separate the center-of-mass-like motion $ c^2 H^{-1} \mathbf{p} t $ from the oscillatory term $ -\frac{i\hbar c^2}{2} H^{-1} \mathbf{p} H^{-1} e^{-2iHt/\hbar} $, identifying the latter as longitudinal zitterbewegung of mass.
• Proposes that charge scanning of electromagnetic fields occurs via zitterbewegung at speed c, contrasting with hypothetical photonic zitterbewegung for gravitational field scanning.

Experimental results

Research questions

• RQ1 Why does the Dirac equation predict electron velocity eigenvalues of ±c, contradicting special relativity and observation?
• RQ2 What is the physical nature of zitterbewegung—does it involve the motion of the electron's mass or its charge?
• RQ3 Can zitterbewegung of the electron's charge produce measurable effects in strong electromagnetic fields?
• RQ4 Is zitterbewegung responsible for the electron's rest mass energy, as suggested by Sakharov and Puthoff?
• RQ5 How does the distinction between charge and mass motion in zitterbewegung affect the interpretation of the electron's internal structure?

Key findings

• The velocity operator $ c\boldsymbol{\alpha} $ corresponds to the velocity of the electron's charge, not its mass, resolving the ±c eigenvalue paradox.
• Zitterbewegung is a real spatial oscillation of the electron's charge at the speed of light, occurring only in superpositions of positive and negative energy states.
• In the supercritical electric field limit $ W \to 0 $, the amplitude of charge zitterbewegung vanishes, and the electron-positron pair separation tends to zero.
• The term $ -\frac{i\hbar c^2}{2} H^{-1} \mathbf{p} H^{-1} e^{-2iHt/\hbar} $ represents a longitudinal zitterbewegung of the electron's mass, but it does not propagate at speed c.
• Zitterbewegung cannot be the source of the electron's rest mass energy, as it is not associated with kinetic energy of massive motion.
• The electron's charge, not its mass, must scan the electromagnetic field at speed c, suggesting a deeper role for zitterbewegung in field interaction.